Theorem carden2b 9877

Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 9876 are meant to replace carden 10459 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
carden2b	⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))

Proof of Theorem carden2b

Step	Hyp	Ref	Expression
1		cardne 9875	. . . . 5 ⊢ ((card‘𝐵) ∈ (card‘𝐴) → ¬ (card‘𝐵) ≈ 𝐴)
2		ennum 9857	. . . . . . . 8 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))
3	2	biimpa 476	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → 𝐵 ∈ dom card)
4		cardid2 9863	. . . . . . 7 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
5	3, 4	syl 17	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐵)
6		ensym 8938	. . . . . . 7 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
7	6	adantr 480	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → 𝐵 ≈ 𝐴)
8		entr 8941	. . . . . 6 ⊢ (((card‘𝐵) ≈ 𝐵 ∧ 𝐵 ≈ 𝐴) → (card‘𝐵) ≈ 𝐴)
9	5, 7, 8	syl2anc 584	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐴)
10	1, 9	nsyl3 138	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → ¬ (card‘𝐵) ∈ (card‘𝐴))
11		cardon 9854	. . . . 5 ⊢ (card‘𝐴) ∈ On
12		cardon 9854	. . . . 5 ⊢ (card‘𝐵) ∈ On
13		ontri1 6349	. . . . 5 ⊢ (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
14	11, 12, 13	mp2an 692	. . . 4 ⊢ ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
15	10, 14	sylibr 234	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) ⊆ (card‘𝐵))
16		cardne 9875	. . . . 5 ⊢ ((card‘𝐴) ∈ (card‘𝐵) → ¬ (card‘𝐴) ≈ 𝐵)
17		cardid2 9863	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
18		id 22	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≈ 𝐵)
19		entr 8941	. . . . . 6 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → (card‘𝐴) ≈ 𝐵)
20	17, 18, 19	syl2anr 597	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) ≈ 𝐵)
21	16, 20	nsyl3 138	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → ¬ (card‘𝐴) ∈ (card‘𝐵))
22		ontri1 6349	. . . . 5 ⊢ (((card‘𝐵) ∈ On ∧ (card‘𝐴) ∈ On) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵)))
23	12, 11, 22	mp2an 692	. . . 4 ⊢ ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵))
24	21, 23	sylibr 234	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ⊆ (card‘𝐴))
25	15, 24	eqssd 3949	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
26		ndmfv 6864	. . . 4 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
27	26	adantl 481	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = ∅)
28	2	notbid 318	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (¬ 𝐴 ∈ dom card ↔ ¬ 𝐵 ∈ dom card))
29	28	biimpa 476	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → ¬ 𝐵 ∈ dom card)
30		ndmfv 6864	. . . 4 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅)
31	29, 30	syl 17	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐵) = ∅)
32	27, 31	eqtr4d 2772	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
33	25, 32	pm2.61dan 812	1 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ⊆ wss 3899  ∅c0 4283   class class class wbr 5096  dom cdm 5622  Oncon0 6315  ‘cfv 6490   ≈ cen 8878  cardccrd 9845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-er 8633  df-en 8882  df-card 9849

This theorem is referenced by:  card1  9878  carddom2  9887  cardennn  9893  cardsucinf  9894  pm54.43lem  9910  nnadju  10106  nnadjuALT  10107  ficardun  10109  ackbij1lem5  10131  ackbij1lem8  10134  ackbij1lem9  10135  ackbij2lem2  10147  carden  10459  r1tskina  10691  cardfz  13891